Position
number ##,

coordinates: #,

as
FR: #,

projected: #.

Occupied
by
root ##,

coordinates: #,

as
FR: #,

projected: #.

Position↔occupied angle: #.

Currently # roots
fixed, # moved by π/3,
# by π/2,
# by 2π/3,
# by π.

Actions: reinitialize, scramble,
Weyl word simplify
(# moves).

Show current Weyl word, set a word, apply a
word. Show cycle
structure.

Apply sample Weyl group elements: of order 8 (centralizer of order
192), of order 8 (centralizer of
order 64), of order 30 (centralizer
of order 30), of order 24
(centralizer of order 24), of order
20 (centralizer of order 20), of
order 18 (centralizer of order 36), of order 14 (centralizer of order
28).

Choose projection: default, alternate, squares, Petrie
(8 30-gons), 10
24-gons, 12 20-gons, 13 (blurred) 18-gons, 7 (blurred) 14-gons; randomly perturb
projection.

Coloring: recolor to
projection, toggle color
scheme, briefly show displacement
angles.

[See
also this
page for a different interactive view of E_{8}, showing
its rotation under the Lie group G_{2}.]

The display above shows a particular projection of
the E_{8} root
system, which can be described as a remarkable polytope in 8
dimensions (also known as
the Gosset
4_{21} polytope) having 240 vertices (known, in this
context, as “roots”), and 6720 edges shown by
black lines in the figure. Quite concretely, the roots can be
described, in the coordinate system we have chosen, as the (112)
points having coordinates (±1,±1,0,0,0,0,0,0) (where both
signs can be chosen independently and the two non-zero coordinates can
be anywhere) together with those (128) having coordinates
(±½,±½,±½,±½,±½,±½,±½,±½)
(where all signs can be chosen independently except that there must be
an even number of minuses).

Every root (being identified with the vector leading from the
origin to the vertex in question) is of the same length (i.e., all
vertices are on a sphere with the origin as center; this is specific
to E_{8} and is not a property of all root systems); the
opposite of each root is again a root, and each one is orthogonal to
126 others, while forming an angle of π/3 with 56 others (those
that are connected to it by an edge): the only possible angles between
two roots are 0, π/3, π/2, 2π/3 and π.

The group of symmetries of this object is the group, known as
the Weyl group
of E_{8}, generated by the (orthogonal) reflections about the
hyperplane orthogonal to each root: this is a group of order 696729600
which can also be described as O_{8}^{+}(2). It is
also the group of automorphism of the adjacency graph of the
polytope.

Those 112 roots which have coordinates of the form
(±1,±1,0,0,0,0,0,0) are shown as larger dots, and constitute
a so-called D_{8} root system inside the E_{8} root
system, which, as a polytope, is
a rectified
octacross; the reflections determined by those vertices generate a
subgroup of order 5160960 (the Weyl group of D_{8}, a subgroup
of index 2 in {±1}≀𝕾_{8}) of the full Weyl
group of E_{8}. The 128 remaining vertices (forming
a demiocteract)
are shown as smaller dots; alone, they are not a root system because
the reflection determined by one of them does not fix that subset.
Note that this division of the 240 vertices as 112+128 is particular
to the chosen coordinate system and is not preserved by symmetries of
the whole (except, precisely, by those living in the smaller Weyl
group of D_{8}; so there are 135 ways of making this
decomposition).

One can further divide the roots in two by calling half of them
“positive” in such a way that the sum of two
positive roots, if it is a root, is always positive, and that for
every root either it or its opposite is positive; there are many ways
to do this (in fact, precisely as many as there are elements in the
Weyl group), and we have chosen the division given by a lexicographic
order on the coordinates: we call positive those roots such that the
leftmost nonzero coordinate is positive (or, by numbering the roots
lexicographically from 0 to 239, the positive ones are those numbered
120 through 239). A choice of positive roots is equivalent to a
choice of fundamental (or simple) roots: these
are the positive roots which cannot be written as a sum of two
positive roots, and it then turns out that these form a basis of the
ambient 8-space and, remarkably, that every positive root can be
written as a linear combination of fundamental roots with nonnegative
integer coefficients (equivalently, the fundamental roots form a
non-orthogonal basis in which the coordinates of every root are either
all nonnegative or all nonpositive; there is a uniquely
defined greatest root, whose coordinates in terms of
fundamental roots dominates that of every other root, and which
happens to be one half the sum of *all* positive roots,
fundamental or not: for E_{8}, it is
⟨4,3,6,5,4,3,2,2⟩ and, for our choices, it is root
number 239, or (1,1,0,0,0,0,0,0)). Any choice of positive/fundamental
roots can be brought to any other choice by a unique element of the
Weyl group.

If we represent the eight fundamental roots and connect two by a
line whenever they form an angle of 2π/3 (the only other
possibility being that they are orthogonal: in the case of
E_{8}, the angles of 3π/4 and 5π/6 do not occur),
we obtain the so-called Dynkin diagram, which in the case
of E_{8} has seven nodes in a simple chain and an eighth
branching from the third. Here, we number the fundamental roots in
the same total order as chosen to define the positive roots (i.e.,
lexicographic order on the coordinates; then the fundamental roots 1
through 8 are the roots numbered 120, 121, 122, 126, 132, 140, 150 and
162), and the Dynkin diagram has fundamental roots
8–1–3–4–5–6–7 in a chain and
fundamental root number 2 branching off from 3.

The fundamental roots are important because the reflection with respect to them suffice to generate the Weyl group. Furthermore, the minimal length of an expression of a given element of the Weyl group as such a product of fundamental reflections (the length relative to the given element for the chosen system of fundamental roots) is equal to the number of positive roots whose image is a negative root; and composing by a fundamental reflection will always increase or decrease by 1 the length of the Weyl group element.

The graphic displays a particular two-dimensional projection of the
eight-dimensional E_{8} root system. Various predefined
projections are possible, emphasizing various symmetries in the Weyl
group. The vertices (roots) are identified by their color and their
size (vertices belonging to the D_{8} root system are shown
larger); hovering the mouse cursor above a vertex shows information
about the corresponding root, both the one which was initially present
at this position and the one which has been placed their by applying
whatever symmetry from the Weyl group: coordinates are given in the
chosen orthonormal system and also as an integer (all nonnegative or
all nonpositive) combination of fundamental roots, as well as the
coordinates in the two-dimensional projection. The angle by which the
root was displaced is also indicated.

Clicking on a root performs the reflection with respect to that
root (it permutes that root with its opposite, fixes 126 others, and
exchanges the 112 remaining roots as 56 pairs). The 696729600
elements of the Weyl group are generated by such reflections, and
determine as many configurations of this widget. Reinitialize

resets every root in its original position, whereas scramble

chooses a random configuration (more or less uniformly in the Weyl
group).

Each element of the Weyl group can be written as a product (of a
uniquely defined length) of reflections by eight fundamental roots:
the simplify

command will perform a single step toward
resolving the word by applying one of these fundamental reflections,
the minimal number of which is displayed (the number of roots which
are in their original place is also displayed).
The show

, set

and apply

commands are used to
display the current Weyl group element as a word of minimal length in
the letters 1 through 8 (indicating the fundamental reflections), set
it to an arbitrary value, or compose it with an arbitrary value: thus,
the simplify

command essentially deletes the last letter in the
word in question. Note that fundamental reflections mean relative to
the *original* position of the fundamental roots, and are
applied left to right in the word order; but it is equivalent to apply
the reflections relative to their displaced position when reading the
word right to left.

The show cycle structure

command displays the number of
cycles of each occurring length, of the current Weyl group element as
acting on the 240 roots. The commands applying various sample Weyl
group elements are given because (except the second element of order
8, which is a cyclic rotation of the coordinates in the chosen basis)
they emphasize the symmetries of each of the predefined
projections.

Most of the predefined projections have a rotational symmetry,
meaning that the root projections are nicely arranged on
regular `n`-gons: for example, one displays the 240 roots as
8 concentric 30-gons, the outer one being a
so-called Petrie
polygon for that polytope; there are, of course, a great many
possible such projections, since one could apply any element of the
Weyl group, but the one proposed here attempts to be very close to the
default projection, so that passing from one projection to another
will not move the vertices in too wild a fashion—however, this
choice has not been made in a very systematic manner. The default
(original) projection is related to the chosen coordinate system in
that it can be described by linearly combining the coordinates with
coefficients given by eight consecutive complex sixteenth roots of
unity. There is also a command which applies a small random
(gaussian) perturbation to the current projection, which gives a feel
of how the root system can be moved slightly.

Note that while changing the projection moves the vertices and
edges along, *it does not change their color* or size: the
vertices remain colored as they were before the projection was
changed. To apply a color scheme harmonious to the chosen projection
(at least in the unscrambled state!), use the recolor to
projection

command. (This is typically what you want if you wish
to visualize a given Weyl group elements in various projections; on
the other hand, to compare the different projections with respect to
each other, try changing projections without recoloring.) There is
also a second color scheme (independent of projection) which can be
used, in which positive roots (for the particular order chosen) are
represented in blue and negative roots in green, and the eight
fundamental roots (relative to that order) are labeled. Finally,
the show displacement angles

will temporarily replace the
coloring of the roots do indicate by what angle they have been
displaced under the current Weyl group element.